arXiv subjects

Francisco Frutos-Alfaro

Publications and source records attributed to Francisco Frutos-Alfaro.

At least 19 recordsLinked to original sources

arXiv2026

Teukolsky Master Equation for a Kerr-like metric with Mass Quadrupole Moment

The Teukolsky Master Equation (TME) is a powerful technique used to study perturbations of a spacetime with gravitational, electromagnetic, and neutrino components. The well-known Kerr metric is a suitable candidate for this formalism because it is a Petrov type D metric. However, the Kerr metric represents the exterior spacetime of a perfect sphere and only black holes are described using this metric. For another astrophysical object, such as fast-spinning neutron stars, the Kerr metric is not valid due to the rotation induced deformations to the gravitational source. The main goal of the present work is to show a TME for a Kerr-like metric (KL) with a mass quadrupole moment parameter $q$. This new parameter will represent deformations from a perfect sphere; therefore, KL is a better candidate to study perturbations surrounding spinning stars.

gr-qc↗

arXiv2026

Chaotic Dynamics due Prolate and Oblate Sources in Kerr-like and Hartle-Thorne Spacetimes with and without Magnetic Field

As demonstrated by observations, every stellar-mass object rotates around some axis; some objects spin faster than others due to different mechanisms. Furthermore, these spinning objects are slightly deformed and are no longer perfect spheres because of hydrostatic equilibrium. The well-known Kerr solution of the Einstein Field Equations (EFE) represents the spacetime surrounding a rotating spherical gravitational source. However, real objects deviate from a perfect sphere and may be prolate or oblate. There are several solutions of the EFE that represent the spacetime of deformed objects. The Kerr--like (KL) metric represents the spacetime surrounding this kind of object, where the deformation is characterized by the mass quadrupole moment parameter $q_{\mathrm{KL}}$. When $q_{\mathrm{KL}} \neq 0$, the Carter constant no longer exists and the equations of motion (EOM) are no longer integrable; therefore, the system exhibits chaotic orbits. Another widely used solution is the Hartle--Thorne (HT) metric, which has similar characteristics and represents a slightly deformed, slowly rotating star. The HT metric has several versions, and two of them were selected to test their validity. The traditional HT version, which contains logarithmic terms, is less accurate than the version with exponential terms. Moreover, both the KL and HT metrics may be extended to include contributions due to the magnetic dipole moment of the source. The equations of motion (EOM) were computed, and these new dynamical systems display several interesting features, which are shown in their Poincar'e sections.

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arXiv2025

On defining astronomically meaningful Reference Frames in General Relativity

In a recent paper we discussed when it is possible to define reference frames nonrotating with respect to distant inertial reference objects (extension of the IAU reference systems to exact general relativity), and how to construct them. We briefly review the construction, illustrating it with further examples, and caution against the recent misuse of zero angular momentum observers (ZAMOs).

gr-qc↗

arXiv2025

Classical general relativity effects by magnetars with massive quadrupole, angular momentum and a magnetic dipole

In this contribution, we obtain classical tests of general relativity using the Hartle-Thorne metric endowed with magnetic dipole and electric charge. This metric represents the approximate stationary spacetime of a massive object with the other characteristics mentioned. These tests are light deflection, time delay, peri-astron precession, and gravitational redshift. We also provide numerical estimates for real magnetars and magnetar candidates from the McGill magnetar catalog, the millisecond pulsar PSR B1257+12 and for the Sun in low-activity cycles. Our results find that, although the magnetic dipole moment contribution tends to be negligible compared to the total amount, its comparison to the massive quadrupole moment and rotational contributions varies from one classical test to the next. For light deflection, the magnetic dipole contribution is about 2 orders of magnitude smaller, compared to the rotational contribution. The magnetic dipole moment contribution is present, but is about 6 orders of magnitude smaller than the second-order rotational contribution to the periastron precession, 5 orders of magnitude smaller for the time delay, and negligible within the approximation presented for the gravitational redshift. The magnetic dipole contribution 1 for the calculations made with PSR B1257+12 was also negligible, but the rotational and quadrupole moment contributions were more significant, which makes the argument for possible future detection stronger than the magnetar case. The rotation, massive quadrupole moment and magnetic dipole contributions for the Sun turned out to be negligible as well.

gr-qc↗

arXiv2025

Innermost Stable Circular Orbits of a Kerr-like Metric with Quadrupole

This paper presents results for the innermost stable circular orbit in a Kerr-like spacetime. The metric employed is an approximation that combines the Kerr metric with the Erez-Rosen metric, expanded in a Taylor series. Consequently, this spacetime incorporates three relativistic multipole moments: mass, spin, and quadrupole moment. Our derivation builds upon the analysis conducted by Chandrasekhar for the Kerr metric. Utilizing the Euler-Lagrange method and Hamiltonian dynamics, we define an effective potential for the radial coordinate. This equation can be used to measure the mass quadrupole through observational methods, as it yields a quadratic polynomial for the quadrupole moment. As anticipated, the limiting cases of this equation correspond to the established cases of Kerr and Schwarzschild spacetimes.

gr-qc↗

arXiv2024

An Approximate Kerr-Newman-like Metric Endowed with a Magnetic Dipole and Mass Quadrupole

Approximate all-terrain spacetimes for astrophysical applications are presented. The metrics possess five relativistic multipole moments, namely mass, rotation, mass quadrupole, charge, and magnetic dipole moment. All these spacetimes approximately satisfy the Einstein-Maxwell field equations. The first metric is generated by means of the Hoenselaers-Perjés method from given relativistic multipoles. The second metric is a perturbation of the Kerr-Newman metric, which makes it a relevant approximation for astrophysical calculations. The last metric is an extension of the Hartle-Thorne metric that is important for obtaining internal models of compact objects perturbatively. The electromagnetic field is calculated using Cartan forms for locally nonrotating observers. These spacetimes are relevant to infer properties of compact objects from astrophysical observations. Furthermore, the numerical implementations of these metrics are straightforward, making them versatile for simulating the potential astrophysical applications.

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arXiv2024

Transparent Spheres as Gravitational Lens

In this contribution, we present a short account of gravitational lenses and how to calculate different properties of its images in the case of having a transparent distribution of matter such as the uniform transparent sphere, isothermal gas sphere, non-singular isothermal gas sphere and a transparent King profile. With the help of XFGLenses software, and numerical methods, different images arising from all of these profiles, and the different caustics and critical curves are shown. The images were consistent with several previous results that are expected for transparent profiles, like having an odd number of images, and reducing the number of images by two when the source passes through the caustic. The curves shown in the caustics where the diamond, the ellipse and the lemniscate-like. For the critical curves, the most common curve was the ellipse, and the lemniscate appeared in the transparent NSIS case, which is consistent with the fact that these curves are common in gravitational lenses.

astro-ph.CO↗

arXiv2024

Equatorial circular motion in a Kerr-like spacetime

An study of the equatorial circular motion of photons and massive particles around a rotating compact body like a neutron star is presented. For this goal, we use an approximate Kerr-like metric with mass quadrupole as perturbation. The effect of this deformation on the photon sphere, and the innermost stable circular orbit, is determined via an effective potential. Furthermore, a stability anaysis is shown, where we observed the same behavior of the effective potential for co-rotating and counter-rotating particles. A comparison with the results for the innermost stable circular orbit obtained for the Hartle-Thorne spacetime is also presented.

gr-qc↗

arXiv2021

Chaotic Behavior of Geodesics in Kerr-like Spacetime

In this contribution, the motion of unitary mass test particles in a perturbed Kerr-like metric is studied using simulations in the configuration and phase space. Our metric represents the approximate exterior spacetime of a massive rotating body with mass quadrupole moment $q$. Chaotic behavior arises as the $q$ parameter increases. The rotation number is determined for an axis of symmetry in the corresponding Poincaré section. The existence of chaotic regions in the region near the source event horizon is shown.

gr-qc↗

arXiv2021

Effects of the treatment of the mass quadrupole moment on ray-tracing applications for rapidly-rotating neutron stars

The Neutron Star Interior Composition Explorer (NICER) mission has provided a unique opportunity to constrain the equation of state of neutron stars by using the technique of pulse-profile modelling. This technique requires accurate and efficient ray tracing, that in turn requires a robust representation of the spacetime around a neutron star. Several exact and approximate metrics have been proposed, and used, to perform ray tracing around neutron stars, with both moderate and fast rotation. In this paper, we perform a comparison between several of these metrics, when used for ray tracing. We calculate the shape of the neutron star as seen by a distant observer using two different surface formulae, the thermal spectrum and pulse profiles from circular and crescent-shaped hotspots, for four configurations of pulsars with rotation rates ranging from 622 to 1000 Hz, and using both a moderate and a stiff equation of state to include realistic and extreme cases. We find small differences between the metrics for rotation frequencies starting at ~700 Hz that could theoretically be used for constraining the quadrupole moment or the spacetime models. We also determine the practicality of use of each metric in larger-scale applications such as pulse-profile

gr-qc↗

arXiv2019

On Relativistic Multipole Moments of Stationary Spacetimes

Among the known exact solutions of Einstein vacuum field equations the Manko-Novikov and the Quevedo-Mashhoon metrics might be suitable ones for the description of the exterior gravitational field of some real non-collapsed body. A new proposal to represent such exterior field is the stationary q-metric. In this contribution, we computed by means of the Fodor-Hoenselaers-Perjes formalism the lowest ten relativistic multipole moments of these metrics. Corresponding moments were derived for the static vacuum solutions of Gutsunayev-Manko and Hernandez-Martin. A direct comparison between the multipole moments of these non-isometric spacetimes is given.

gr-qc↗

arXiv2019

Approximate Spacetime for Neutron Stars

An approximate realistic metric representing the spacetime of neutron stars is obtained by perturbing the Kerr metric. This metric has five parameters, namely the mass, spin or angular momentum, mass quadrupole, spin octupole and mass hexadecapole. Moreover, a version of the Hartle-Thorne metric containing these parameters is constructed by means of a series transformation between these spacetimes and solving the Einstein field equations. The form of the Pappas metric in Schwarzschild spherical coordinates is found. The three relativistic multipole structures are compared.

gr-qc↗

arXiv2018

On the Post-linear Quadrupole-Quadrupole Metric

The Hartle-Thorne metric defines a reliable spacetime for most astrophysical purposes, for instance for the simulation of slowly rotating stars. Solving the Einstein field equations, we added terms of second order in the quadrupole moment to its post-linear version in order to compare it with solutions found by Blanchet in the frame of the multi-polar post-Minkowskian framework. We first derived the extended Hartle-Thorne metric in harmonic coordinates and then showed agreement with the corresponding post-linear metric from Blanchet. We also found a coordinate transformation from the post-linear Erez-Rosen metric to our extended Hartle-Thorne spacetime. It is well known that the Hartle-Thorne solution can be smoothly matched with an interior perfect fluid solution with physically appropriate properties. A comparison among these solutions provides a validation of them. It is clear that in order to represent realistic solutions of self-gravitating (axially symmetric) matter distributions of perfect fluid, the quadrupole moment has to be included as a physical parameter.

gr-qc↗

arXiv2018

Approximate Kerr-like Metric with Quadrupole

A new approximate metric representing the spacetime of a rotating deformed body is obtained by perturbing the Kerr metric to include til the second order of the quadrupole moment. It has a simple form, because is Kerr-like. Its Taylor expansion form coincides with second order quadrupole metrics with slow rotation already found. Moreover, it can be transformed to an improved Hartle-Thorne metric, this guarantees its validity to be useful in studying compact object, and that it is possible to find an inner solution.

gr-qc↗

arXiv2017

A numerical study of approximate metrics with quadrupole moment

Recently, spacetimes described by metrics with three parameters (mass, rotation and small quadrupole moment) was found, and in this paper, null geodesics for these metrics are calculated and visualized. Light scattering, as well as the role that the quadrupole moment plays in deforming these kinds of spacetimes are studied. This is a new application of a symbolic--numeric program that we previously used to study the Bonnor metric.

gr-qc↗

arXiv2017

Innermost stable circular orbits of a Kerr-like Metric with Quadrupole

The innermost stable circular orbit equation of a test particle is obtained for an approximate Kerr-like spacetime with quadrupole moment. We derived the effective potential for the radial coordinate by the Euler-Lagrange method. This equation can be employed to measure the mass quadrupole by observational means, because from this equation a quadratic polynomial for the quadrupole moment can be found. As expected, the limiting cases of this equation are found to be the known cases of Kerr and Schwarzschild.

gr-qc↗

arXiv2017

Comparison of Vacuum Static Quadrupolar Metrics

We investigate the properties of static and axisymmetric vacuum solutions of Einstein equations which generalize the Schwarzschild spherically symmetric solution to include a quadrupole parameter. We test all the solutions with respect to elementary and asymptotic flatness and curvature regularity. Analyzing their multipole structure, according to the relativistic invariant Geroch definition, we show that all of them are equivalent up to the level of the quadrupole. We conclude that the $q-$metric, a variant of the Zipoy-Voorhees metric, is the simplest generalization of the Schwarzschild metric, containing a quadrupole parameter.

gr-qc↗

arXiv2016

Approximate Kerr-Newman-like Metric with Quadrupole

The Kerr metric is known to present issues when trying to find an interior solution. In this work we continue in our efforts to construct a more realistic exterior metric for astrophysical objects. A new approximate metric representing the spacetime of a charged, rotating and slightly-deformed body is obtained by perturbing the Kerr-Newman metric to include the mass-quadrupole and quadrupole-quadrupole orders. It has a simple form, because is Kerr-Newman-like. Its post-linear form without charge coincides with post-linear quadrupole-quadrupole metrics already found.

gr-qc↗